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Constructions of k-regular maps using finite local schemes | | Jarosław Buczyński
; Tadeusz Januszkiewicz
; Joachim Jelisiejew
; Mateusz Michałek
; | | Date: |
18 Nov 2015 | | Abstract: | A continuous map $R^m o R^N$ or $C^m o C^N$ is called $k$-regular if the
images of any $k$ points are linearly independent. Given integers $m$ and $k$ a
problem going back to Chebyshev and Borsuk is to determine the minimal value of
$N$ for which such maps exist. The methods of algebraic topology provide lower
bounds for $N$, however there are very few results on the existence of such
maps for particular values $m$. Using the methods of algebraic geometry we
construct $k$-regular maps. We relate the upper bounds on $N$ with the
dimension of the locus of certain Gorenstein schemes in the punctual Hilbert
scheme. The computations of the dimension of this family is explicit for $kleq
9$, and we provide explicit examples for $kleq 5$. We also provide upper
bounds for arbitrary $m$ and $k$. | | Source: | arXiv, 1511.5707 | | Services: | Forum | Review | PDF | Favorites | |
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